An optimal decay estimate for the linearized water wave equation in 2D

We obtain a decay estimate for solutions to the linear dispersive equation $iu_t-(-\Delta)^{1/4}u=0$ for $(t,x)\in\mathbb{R}\times\mathbb{R}$. This corresponds to a factorization of the linearized water wave equation $u_{tt}+(-\Delta)^{1/2}u=0$. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order $|t|^{-1/2}$ for solutions corresponding to data $u(0)=\varphi$, assuming only bounds on $\lVert \varphi\rVert_{H_x^1(\mathbb{R})}$ and $\lVert x\partial_x\varphi\rVert_{L_x^2(\mathbb{R})}$. As another application of these ideas, we give an extension to equations of the form $iu_t-(-\Delta)^{\alpha/2}u=0$ for a wider range of $\alpha$.Comment: New result added (see Section 3). To appear in Proc. Amer. Math. So

The defocusing energy-supercritical cubic nonlinear wave equation in dimension five

We consider the energy-supercritical nonlinear wave equation $u_{tt}-\Delta u+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no radial assumption on the initial data. We prove that a uniform-in-time {\it a priori} bound on the critical norm implies that solutions exist globally in time and scatter at infinity in both time directions. Together with our earlier works in dimensions $d\geq 6$ with general data and dimension $d=5$ with radial data, the present work completes the study of global well-posedness and scattering in the energy-supercritical regime for the cubic nonlinearity under the assumption of uniform-in-time control over the critical norm.Comment: AMS Latex, 45 pages. Final versio

Negative energy blowup results for the focusing Hartree hierarchy via identities of virial and localized virial type

We establish virial and localized virial identities for solutions to the Hartree hierarchy, an infinite system of partial differential equations which arises in mathematical modeling of many body quantum systems. As an application, we use arguments originally developed in the study of the nonlinear Schr\"odinger equation (see work of Zakharov, Glassey, and Ogawa--Tsutsumi) to show that certain classes of negative energy solutions must blow up in finite time. The most delicate case of this analysis is the proof of negative energy blowup without the assumption of finite variance; in this case, we make use of the localized virial estimates, combined with the quantum de Finetti theorem of Hudson and Moody and several algebraic identities adapted to our particular setting. Application of a carefully chosen truncation lemma then allows for the additional terms produced in the localization argument to be controlled.Comment: 25 pages, final versio

Maximizers for the Strichartz Inequalities for the Wave Equation

We prove the existence of maximizers for Strichartz inequalities for the wave equation in dimensions $d\geq 3$. Our approach follows the scheme given by Shao, which obtains the existence of maximizers in the context of the Schr\"odinger equation. The main tool that we use is the linear profile decomposition for the wave equation which we prove in $\mathbb{R}^d$, $d\geq 3$, extending the profile decomposition result of Bahouri and Gerard, previously obtained in $\mathbb{R}^3$.Comment: 28 pages, revised version, minor change

Gibbs measure evolution in radial nonlinear wave and Schr\"odinger equations on the ball

We establish new results for the radial nonlinear wave and Schr\"odinger equations on the ball in $\Bbb R^2$ and $\Bbb R^3$, for random initial data. More precisely, a well-defined and unique dynamics is obtained on the support of the corresponding Gibbs measure. This complements results from \cite{B-T1,B-T2} and \cite {T1,T2}.Comment: Research announcement, 6 page

Almost sure global well posedness for the radial nonlinear Schrodinger equation on the unit ball I: the 2D case

Our first purpose is to extend the results from \cite{T} on the radial defocusing NLS on the disc in $\mathbb{R}^2$ to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs measure (which is the natural extension of the classical flow for smooth data). We follow a similar approach as in \cite{BB-1} exploiting certain additional a priori space-time bounds that are provided by the invariance of the Gibbs measure. Next, we consider the radial focusing equation with cubic nonlinearity (the mass-subcritical case was studied in \cite{T2}) where the Gibbs measure is subject to an $L^2$-norm restriction. A phase transition is established, of the same nature as studied in the work of Lebowitz-Rose-Speer \cite{LRS} on the torus. For sufficiently small $L^2$-norm, the Gibbs measure is absolutely continuous with respect to the free measure, and moreover we have a well-defined dynamics.Comment: 25 page

Invariant Gibbs measure evolution for the radial nonlinear wave equation on the 3D ball

We establish new global well-posedness results along Gibbs measure evolution for the nonlinear wave equation posed on the unit ball in $\mathbb{R}^3$ via two distinct approaches. The first approach invokes the method established in the works \cite{B1,B2,B3} based on a contraction-mapping principle and applies to a certain range of nonlinearities. The second approach allows to cover the full range of nonlinearities admissible to treatment by Gibbs measure, working instead with a delicate analysis of convergence properties of solutions. The method of the second approach is quite general, and we shall give applications to the nonlinear Schr\"odinger equation on the unit ball in subsequent works \cite{BB1,BB2}.Comment: 22 page